Stokes flow equations have been implemented successfully in practice for simulating problems with moving interfaces. Though computational methods produce accurate solutions and numerical convergence can be demonstrated using a resolution study, the rigorous convergence proofs are usually limited to particular reformulations and boundary conditions. In this paper, a rigorous error analysis of the marker and cell (MAC) scheme for Stokes interface problems with constant viscosity in the framework of the finite difference method is presented. Without reformulating the problem into elliptic PDEs, the main idea is to use a discrete Ladyzenskaja-Babuska-Brezzi (LBB) condition and construct auxiliary functions, which satisfy discretized Stokes equations and possess at least second order accuracy in the neighborhood of the moving interface. In particular, the method, for the first time, enables one to prove second order convergence of the velocity gradient in the discrete $\ell^2$-norm, in addition to the velocity and pressure fields. Numerical experiments verify the desired properties of the methods and the expected order of accuracy for both two-dimensional and three-dimensional examples.
In this paper, we develop the numerical theory of decoupled modified characteristic finite element method with different subdomain time steps for the mixed stabilized formulation of nonstationary dual-porosity-Navier-Stokes model. Based on partitioned time-stepping methods, the system is decoupled, which means that the Navier-Stokes equations and two different Darcy equations are solved independently at each time step of subdomain. In particular, the Navier-Stokes equations are solved by the modified characteristic finite element method, which overcome the computational difficulties caused by …
For the discretization of the convective term in the Navier-Stokes equations (NSEs), the commonly used convective formulation (CONV) does not preserve the energy if the divergence constraint is only weakly enforced. In this paper, we apply the skew-symmetrization technique in [B. Cockburn, G. Kanschat and D. Schötzau, Math. Comp., 74 (2005), pp. 1067-1095] to conforming finite element methods, which restores energy conservation for CONV. The crucial idea is to replace the discrete advective velocity with its a $H(\operatorname{div})$-conforming divergence-free approximation in CONV. We prove t…
A discontinuous viscosity coefficient makes the jump conditions of the velocity and normal stress coupled together, which brings great challenges to some commonly used numerical methods to obtain accurate solutions. To overcome the difficulties, a kernel free boundary integral (KFBI) method combined with a modified marker-and-cell (MAC) scheme is developed to solve the two-phase Stokes problems with discontinuous viscosity. The main idea is to reformulate the two-phase Stokes problem into a single-fluid Stokes problem by using boundary integral equations and then evaluate the boundary integral…
A Strongly Consistent Finite Difference Scheme for Steady Stokes Flow and its Modified Equations
Saratov State University（萨马拉国立大学）；Joint Institute for Nuclear Research（联合核子研究所）；Peoples’ Friendship University of Russia（俄罗斯人民友谊大学）；King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）
The need for accelerating the repeated solving of certain parametrized systems motivates the development of more efficient reduced order methods. The classical reduced basis method is popular due to an offline-online decomposition and a mathematically rigorous {\em a posterior} error estimator which guides a greedy algorithm offline. For nonlinear and nonaffine problems, hyper reduction techniques have been introduced to make this decomposition efficient. However, they may be tricky to implement and often degrade the online computation efficiency. To avoid this degradation, reduced residual re…
We present new error estimates for the finite volume and finite difference methods applied to the compressible Navier-Stokes equations. The main innovative ingredients of the improved error estimates are a refined consistency analysis combined with a continuous version of the relative energy inequality. Consequently, we obtain better convergence rates than those available in the literature so far. Moreover, the error estimates hold in the whole physically relevant range of the adiabatic coefficient.
In this paper, a parallel domain decomposition method is proposed for solving the fully-mixed Stokes-dual-permeability fluid flow model with Beavers-Joseph (BJ) interface conditions. Three Robin-type boundary conditions and a modified weak formulation are constructed to completely decouple the original problem, not only for the free flow and dual-permeability regions but also for the matrix and microfractures in the dual-porosity media. We derive the equivalence between the original problem and the decoupled systems with some suitable compatibility conditions, and also demonstrate the equivale…
Error analysis of projection methods for non inf-sup stable mixed finite elements. The Navier-Stokes equations
Universidad de Valladolid（巴利亚多利德大学）；Universidad de Sevilla（塞维利亚大学）；Universidad Autónoma de Madrid（马德里自治大学）；Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM（CSIC-UAM-UC3M-UCM数学科学研究所）
In this work, we propose a local Fourier analysis for multigrid methods with coarsening by a factor of three for the staggered finite-difference method applied to the Stokes equations. In [21], local Fourier analysis has been applied to a mass-based Braess-Sarazin relaxation, a mass-based $σ$-Uzawa relaxation, and a mass-based distributive relaxation, with standard coarsening on staggered grids for the Stokes equations. Here, we consider multigrid methods with coarsening by three for these relaxation schemes. We derive theoretically optimal smoothing factors for this coarsening strategy. The o…
Error analysis of projection methods for non inf-sup stable mixed finite elements. The transient Stokes problem
Universidad de Valladolid（瓦拉多利德大学）；Universidad de Sevilla（塞维利亚大学）；Universidad Autónoma de Madrid（马德里自治大学）；Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM（CSIC-UAM-UC3M-UCM数学科学研究所）
The present paper is focused on the proof of the convergence of the discrete implicit Marker-and-Cell (MAC) scheme for time-dependent Navier--Stokes equations with variable density and variable viscosity. The problem is completed with homogeneous Dirichlet boundary conditions and is discretized according to a non-uniform Cartesian grid. A priori-estimates on the unknowns are obtained, and along with a topological degree argument they lead to the existence of a solution of the discrete scheme at each time step. We conclude with the proof of the convergence of the scheme toward the continuous pr…
Convergence of the MAC scheme for the compressible stationary Navier-Stokes equations
Aix Marseille Univ, CNRS, Centrale Marseille, I2M（艾克斯-马赛大学，法国国家科学研究中心，中央马赛学院）；Institut de Radioprotection et de Sûreté Nucléaire (IRSN)（核辐射防护与安全研究所）；IMATH, Université du Sud Toulon-Var（瓦尔南部大学）
An efficient numerical scheme based on the scalar auxiliary variable (SAV) and marker and cell scheme (MAC) is constructed for the Navier-Stokes equations. A particular feature of the scheme is that the nonlinear term is treated explicitly while being unconditionally energy stable. A rigorous error analysis is carried out to show that both velocity and pressure approximations are second-order accurate in time and space. Numerical experiments are presented to verify the theoretical results.
We prove that some time Euler schemes for the 3D Navier-Stokes equations modified by adding a Brinkman-Forchheimer term and a random perturbation converge in $L^2(Ω)$. This extends previous results concerning the strong rate of convergence of some time discretization schemes for the 2D Navier Stokes equations. Unlike the 2D case, our proposed 3D model with the Brinkman-Forchheimer term allows for a strong rate of convergence of order almost 1/2, that is independent of the viscosity parameter.
The velocity solution of the incompressible Stokes equations is not affected by changes of the right hand side data in form of gradient fields. Most mixed methods do not replicate this property in the discrete formulation due to a relaxation of the divergence constraint which means that they are not pressure-robust. A recent reconstruction approach for classical methods recovers this invariance property for the discrete solution, by mapping discretely divergence-free test functions to exactly divergence-free functions in the sense of $\boldsymbol{H}(\operatorname{div})$. Moreover, the Stokes s…
Robust superconvergence analysis of physics-preserving RMAC scheme for the Stokes and Navier--Stokes equations on non-uniform grids at high Reynolds numbers
Most classical finite element schemes for the (Navier-)Stokes equations are neither pressure-robust, nor are they inf-sup stable on general anisotropic triangulations. A lack of pressure-robustness may lead to large velocity errors, whenever the Stokes momentum balance is dominated by a strong and complicated pressure gradient. It is a consequence of a method, which does not exactly satisfy the divergence constraint. However, inf-sup stable schemes can often be made pressure-robust just by a recent, modified discretization of the exterior forcing term, using $\mathbf{H}(\operatorname{div})$-co…
In this paper, we apply discontinuous finite element Galerkin method to the time-dependent $2D$ incompressible Navier-Stokes model. We derive optimal error estimates in $L^\infty(\textbf{L}^2)$-norm for the velocity and in $L^\infty(L^2)$-norm for the pressure with the initial data $\textbf{u}_0\in \textbf{H}_0^1\cap \textbf{H}^2$ and the source function $\textbf{f}$ in $L^\infty(\textbf{L}^2)$ space. These estimates are established with the help of a new $L^2$-projection and modified Stokes operator on appropriate broken Sobolev space and with standard parabolic or elliptic duality arguments.…
A novel notion for constructing a well-balanced scheme - a gradient-robust scheme - is introduced and a showcase application for a steady compressible, isothermal Stokes equations is presented. Gradient-robustness means that arbitrary gradient fields in the momentum balance are well-balanced by the discrete pressure gradient - if there is enough mass in the system to compensate the force. The scheme is asymptotic-preserving in the sense that it degenerates for low Mach numbers to a recent inf-sup stable and pressure-robust discretization for the incompressible Stokes equations. The convergence…
This article aims at developing a high order pressure-based solver for the solution of the 3D compressible Navier-Stokes system at all Mach numbers. We propose a cell-centered discretization of the governing equations that splits the fluxes into a fast and a slow scale part, that are treated implicitly and explicitly, respectively. A novel semi-implicit discretization is proposed for the kinetic energy as well as the enthalpy fluxes in the energy equation, hence avoiding any need of iterative solvers. The implicit discretization yields an elliptic equation on the pressure that can be solved fo…
Low Mach number limit of a pressure correction MAC scheme for compressible barotropic flows
Aix-Marseille Université, CNRS, École Centrale de Marseille（艾克斯-马赛大学，法国国家科学研究中心，马赛中央理工学院）；Institut de Radioprotection et de Sûreté Nucléaire (IRSN)（核辐射防护与安全研究所）；Université de Lyon, CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan（里昂大学，法国国家科学研究中心UMR 5208，里昂第一大学，卡米尔·若尔当研究所）
